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Solutions from a student (dragged) 32

# Solutions from a student (dragged) 32 - Part(c To get two...

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Part (b): We now want to count the number of n -digit numbers where the digit 0 appears i times. Lets pick the locations where we want to place the zeros. This can be done in n i ways. We then have nine choices for the other digits to place in the other n - i locations. This gives 9 n - i possible enoumerations for non-zero digits. In total then we have n i 9 n - i , n digit numbers with i zeros in them. Problem 9 (selecting three students from three classes) Part (a): To choose three students from 3 n total students can be done in 3 n 3 ways. Part (b): To pick three students from the same class we must first pick the class to draw the student from. This can be done in 3 1 = 3 ways. Once the class has been picked we have to pick the three students in from the n in that class. This can be done in n 3 ways. Thus in total we have 3 n 3 , possible selections of three students all from one class.
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Unformatted text preview: Part (c): To get two students in the same class and another in a di±erent class, we must Frst pick the class from which to draw the two students from. This can be done in ± 3 1 ² = 3 ways. Next we pick the other class from which to draw the singleton student from. Since there are two possible classes to select this student from this can be done in two ways. Once both of these classes are selected we pick the individual two and one students from their respective classes in ± n 2 ² and ± n 1 ² ways respectively. Thus in total we have 3 · 2 · ± n 2 ²± n 1 ² = 6 n n ( n-1) 2 = 3 n 2 ( n-1) , ways. Part (d): Three students (all from a di±erent class) can be picked in ± n 1 ² 3 = n 3 ways. Part (e): As an identity we have then that ± 3 n 3 ² = 3 ± n 3 ² + 3 n 2 ( n-1) + n 3 ....
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